It was the second day of protests in the former Soviet state.
Russia's Interfax news agency quoted Grigory Karasin, Russia's deputy foreign minister, as saying: "We are following the situation with concern."
Javier Solana, the European Union foreign policy chief, urged all sides to refrain from violence.
He said: "I am very concerned over the situation in Chisinau. I call on all sides to refrain from violence and provocation.
"Violence against government buildings is unacceptable. Equally important is the respect for the inalienable right of assembly of peaceful demonstrators."
Earlier, protesters carrying Moldovan and European flags and shouting anti-Communist slogans gathered outside the government buildings before making their way down Chisinau's main boulevard to the president's office.
"The election was controlled by the Communists, they bought everyone off," Alexei, a student, said.
"We will have no future under the Communists because they just think of themselves."
Aisha Jung, a human rights worker in Chisinau, told Al Jazeera: "It has been the most extraordinary morning.
"It started off with a peaceful demonstration in the central square. Everyone then walked peacefully to the presidential and parliamentary building.
"After about half-an-hour or an hour there they started to storm the presidential building with only a few policemen guarding it.
"They then went up the steps of the parliamentary building as well."
Jing said that riot police were brought in, but were outnumbered by the protesters who threw objects at the presidential building.
"It is a protest against the victory of the Communist government. I don't think the people expected the victory to be as outright as it was"
human rights worker
The police then managed to separate the crowd either side of the presidential building by bringing in reinforcements, but then the crowd refused to leave the area.
"It is a protest against the victory of the Communist government. I don't think the people expected the victory to be as outright as it was."
Vladimir Voronin, the president and leader of the Communist party, is not allowed to serve a third consecutive term, but has indicated that he wants to maintain control of the government after the election victory.
It is not clear what position he would take in the new administration.
The country's central election committee said on Monday that the Communists had secured about 50 per cent of the vote in Sunday's polls, with 98 per cent of the ballots counted.
They gained 61 parliamentary seats meaning that they do not need to consult parliament to choose a new president.